Norm-convergence of a linear operator in a Banach space Let $E$ be a Banach space.
The definition of a cone in many in textbooks dealing with order in Banach space is:

A nonempty closed subset $K$ of $E$ is called a convex cone if:

*

*$K\neq \{0_E\}$.

*$K+\lambda K\subseteq K,\;\:\lambda \geq 0$.

*$K\cap -K=\{0_E\}$, that is if $x\in K$ and $-x\in K$, then $x=0_E$.


and  $x \preceq y$ (in $E$) if and only if $y-x \in K$.
Now let we denote by $\mathcal{L}^{*}(E)$  the set of elements $A\in\mathcal{L}(E)$ satisfying the following conditions:

*

*$A K \subseteq K$ (that is $A$ is nondecreasing, i.e., $u \preceq v$ implies $A(u) \preceq A(v)$,

*For all $u \in K,\left\|A^{n} u\right\|_{E} \rightarrow 0$ as $n \rightarrow \infty$.

How do I construct an example of an operator in $\mathcal{L}^{*}(E)$?

I'm thinking about something like this, with the monotony condition.
 A: As @sein mentioned in his comment it depends on what you want. For a specific Banach space, consider the convex cone of nonnegative functions as a subset of $L^2(\mathbb{R})$. Define the operator
$$
A f:= S_1 f \, ,
$$
where $S_1$ is the heat semigroup at $t\geq 1$, i.e. we have $S_1 f= \varphi_1* f$ with $\varphi_1$ the heat kernel at time $1$.  Clearly, $A \in \mathcal{L}^*$, since $\varphi_1$ is nonnegative. Additionally,
\begin{align}
\lVert A^n f\rVert_{L^2(\mathbb{R})}= \lVert \varphi_n * f\rVert_{L^2(\mathbb{R})} \stackrel{n \to \infty}{\to} 0 \, ,
\end{align}
where we simply use the fact that the heat equation converges to $0$ for all $L^2(\mathbb{R})$ initial data. I am sure there are even simpler examples than this.
Edit: On second thought here is a simpler example in the general setting; please correct me if I am making some stupid mistake. Set
$$
A f= \lambda f \, ,
$$
for $\lambda \in (0,1)$. If $u \in K$, then $\lambda u = 0_E + \lambda u \in K$. The convergence to $0$ is straightforward.
